CAPD::DynSys Library  6.0.0
CAPD::DynSys Library Documentation

CAPD is an acronym for Computer Assisted Proofs in Dynamics. The library was initiated in early 1990s by Marian Mrozek as the tool for the computer assisted proof of chaotic dynamics in the Lorenz system recognized by Encyclopædia Britannica as one of the four most important achievements in mathematics in 1995. Over three decades we have developed and implemented new algorithms for rigorous numerical analysis of dynamical systems which constitute the core of the present version of the CAPD::DynSys library.

It is a collection of flexible C++ modules which are mainly designed for nonrigorous and validated numerics for dynamical systems. The library proved to be an efficient tool in rigorous numerical study of various phenomena in dynamical systems, just to mention

  • existence and stability of periodic trajectories
  • the existence of chaotic dynamics in various ODEs and PDEs.
  • the existence of connecting orbits to equlibria and periodic orbits
  • local and global bifurcations (Shilnikov orbits, Bykov cycles, homoclinic tangencies, attractor templates)
  • study of invariant tori for maps and flows
  • existence and hyperbolicity of attractors for ODEs
  • Arnold diffusion

Most important features of the CAPD::DynSys library

The present version of the CAPD::DynSys library implements in particular algorithms for computation of guaranteed bounds for

  • higher order derivatives of multivariate maps
  • solutions to ODEs and associated variational equations
  • solutions to differential inclusions
  • Poincar√© maps and their derivatives
  • solutions to come class of dissipative PDEs

and many others.

Reference

The basic interface of the library is presented in the article

T. Kapela, M. Mrozek, D. Wilczak, P. Zgliczynski,
CAPD::DynSys: a flexible C++ toolbox for rigorous numerical analysis of dynamical systems,
Communications in Nonlinear Science and Numerical Simulation, Volume 101, October 2021, 105578.
doi.org/10.1016/j.cnsns.2020.105578

We also recommend to look at

  • Getting started
    where the interface of the library is introduced step by step and supported by numerous examples.
  • Examples
    for more detailed description, advanced examples and a few complete computer-assisted proofs of various dynamical phenomena.

Download and use the CAPD::DynSys library